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Minimax Analysis of Active Learning

Steve Hanneke, Liu Yang; 16(109):3487−3602, 2015.

Abstract

This work establishes distribution-free upper and lower bounds on the minimax label complexity of active learning with general hypothesis classes, under various noise models. The results reveal a number of surprising facts. In particular, under the noise model of Tsybakov (2004), the minimax label complexity of active learning with a VC class is always asymptotically smaller than that of passive learning, and is typically significantly smaller than the best previously-published upper bounds in the active learning literature. In high-noise regimes, it turns out that all active learning problems of a given VC dimension have roughly the same minimax label complexity, which contrasts with well-known results for bounded noise. In low-noise regimes, we find that the label complexity is well-characterized by a simple combinatorial complexity measure we call the star number. Interestingly, we find that almost all of the complexity measures previously explored in the active learning literature have worst-case values exactly equal to the star number. We also propose new active learning strategies that nearly achieve these minimax label complexities.

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